We constrain a broad class of teleportation protocols using insights from locality. In the "standard" teleportation protocols we consider, all outcome-dependent unitaries are Pauli operators conditioned on linear functions of the measurement outcomes. We find that all such protocols involve preparing a "resource state" exhibiting symmetry-protected topological (SPT) order with Abelian protecting symmetry $\mathcal{G}_{k}= (\mathbb{Z}_2 \times \mathbb{Z}_2)^k$. The $k$ logical states are teleported between the edges of the chain by measuring the corresponding $2k$ string order parameters in the bulk and applying outcome-dependent Paulis. Hence, this single class of nontrivial SPT states is both necessary and sufficient for the standard teleportation of $k$ qubits. We illustrate this result with several examples, including the cluster state, variants thereof, and a nonstabilizer hypergraph state.

中文翻译：

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量子隐形传态意味着对称保护的拓扑顺序


我们使用来自当地的见解来限制一大类传送协议。在我们考虑的 “标准” 隐形传态协议中，所有结果依赖性的幺正都是以测量结果的线性函数为条件的 Pauli 算子。我们发现，所有这些协议都涉及准备一个“资源状态”，该“资源状态”表现出对称保护拓扑 （SPT） 顺序，具有阿贝尔保护对称性 $\mathcal{G}_{k}= （\mathbb{Z}_2 \times \mathbb{Z}_2）^k$。通过批量测量相应的 $2k$ 字符串顺序参数并应用依赖于结果的 Paulis，$k$ 逻辑状态在链的边缘之间传送。因此，这类非平凡的 SPT 状态对于 $k$ 量子比特的标准隐形传态既必要又足够。我们用几个例子来说明这个结果，包括集群状态、其变体和非稳定器超图状态。